A box weighs $196 \;\mathrm{N}$ on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to ....... $N$

(Take $\mathrm{g}=10\; \mathrm{ms}^{-2}$ at the north pole and the radius of the earth $=6400\; \mathrm{km}$)

Acceleration due to gravity on moon is $\frac 16$ of the acceleration due to gravity on earth. If the ratio of densities of earth $({\rho _e})$ and moon $({\rho _m})$ is $\left( {\frac{{{\rho _e}}}{{{\rho _m}}}} \right) = \frac{5}{3}$ then radius of moon $R_m$ in terms of $R_e$ will be

What should be the angular speed with which the earth have to rotate on its axis so that a person on the equator would weigh $\frac{3}{5}$ th as much as present?

If both the mass and the radius of the earth decrease by $1\%$, the value of the acceleration due to gravity will

At what depth below the surface of the earth, acceleration due to gravity $g$ will be half its value $1600 \,km$ above the surface of the earth

A particle of mass $10\, g$ is kept of the surface of a uniform sphere of mass $100\, kg$ and a radius of $10\, cm .$ Find the work to be done against the gravitational force between them to take the particle far away from the sphere. (you make take $\left.G=6.67 \times 10^{-11} Nm ^{2} / kg ^{2}\right)$